paper

Cluster algebras of type

arXiv:0904.2543 · doi:10.1007/s10468-011-9275-5

Abstract

In this paper we study cluster algebras $\myAA$ of type . We solve the recurrence relations among the cluster variables (which form a T--system of type ). We solve the recurrence relations among the coefficients of $\myAA$ (which form a Y--system of type ). In $\myAA$ there is a natural notion of positivity. We find linear bases $\BB$ of $\myAA$ such that positive linear combinations of elements of $\BB$ coincide with the cone of positive elements. We call these bases \emph{atomic bases} of $\myAA$. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type . Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of $\BB$ are parameterized by $\ZZ^3$ via their --vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\myAA$ restricts to a bijection between $\BB$ and $\ZZ^3$. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type . We find explicit recurrence relations to express every element of $\myAA$ as linear combinations of elements of $\BB$.

Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 2011

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